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ARTICLE
OPEN
Differential dynamic microscopy of bidisperse colloidal suspensions Mohammad S. Safari1, Ryan Poling-Skutvik1, Peter G. Vekilov1 and Jacinta C. Conrad
1
Research tasks in microgravity include monitoring the dynamics of constituents of varying size and mobility in processes such as aggregation, phase separation, or self-assembly. We use differential dynamic microscopy, a method readily implemented with equipment available on the International Space Station, to simultaneously resolve the dynamics of particles of radius 50 nm and 1 μm in bidisperse aqueous suspensions. Whereas traditional dynamic light scattering fails to detect a signal from the larger particles at low concentrations, differential dynamic microscopy exhibits enhanced sensitivity in these conditions by accessing smaller wavevectors where scattering from the large particles is stronger. Interference patterns due to scattering from the large particles induce non-monotonic decay of the amplitude of the dynamic correlation function with the wavevector. We show that the position of the resulting minimum contains information on the vertical position of the particles. Together with the simple instrumental requirements, the enhanced sensitivity of differential dynamic microscopy makes it an appealing alternative to dynamic light scattering to characterize samples with complex dynamics. npj Microgravity (2017)3:21 ; doi:10.1038/s41526-017-0027-7
INTRODUCTION Microgravity provides a unique environment in which to investigate the physics of transport processes such as diffusion, convection, and conduction. These processes affect structure in systems featuring sub-microscale constituents, including bacterial biofilms,1, 2 protein crystals,3, 4 and complex fluids.5 Monitoring dynamics on these length scales in microgravity is expected to generate fundamental insight into the physics controlling structural evolution. One traditional method for characterizing dynamics at the microscale, dynamic light scattering (DLS),6 is already available on the International Space Station (ISS)7 but is restricted by the detector frame rate to characterize slow motions. An intriguing alternative is provided by recent enhancements to the Light Microscopy Module (LMM) on the ISS, which increased the time resolution of image acquisition and imparted confocal imaging capabilities. These advances make it possible to access faster dynamics across a broad range of samples but require methods to obtain dynamics from microscopy time series images. For microscale structures that can be directly visualized on an optical microscope, dynamics can be extracted from a time-series of microscopy images via image-processing algorithms. The numerical aperture on a standard light microscope, however, limits the size of resolvable structures to typically greater than 150 nm using the smallest feasible wavelength of light. Although advanced super-resolution microscopy methods can in principle lower this size limit significantly,8 low time resolution and stringent instrumentation requirements limit their immediate adoption on the ISS. Hence there remains a need for methods that can be readily implemented on a standard microscope with relatively simple equipment, compatible with the strict demands of the space environment. Differential dynamic microscopy (DDM),9 an extension of heterodyne near-field scattering10 and one of an emerging family
of digital Fourier techniques,11 is a flexible, powerful, and readilyimplemented method to probe microscale dynamics. In DDM, the microscale dynamics of a sample are extracted from the decorrelation of intensity fluctuations evaluated from a time series of difference images.12 This method has two key advantages: first, it has minimal instrumentation requirements, and, second, it can access smaller wavevectors and hence larger length scales than conventional DLS setups. Thus, DDM has been used to characterize the dynamics of dispersed nanoparticles13–15 and bacteria,16, 17 as well as colloidal18 and protein19 condensates. Further, DDM has been extended to imaging modes beyond brightfield, including fluorescence,12 confocal,20 and darkfield21 microscopy. Hence this method offers new flexibility and capability to investigate complex dynamic phenomena using microscopy. The simple instrumental requirements of DDM allow it to be implemented on the ISS to enable novel probes of dynamics in microgravity. As one example, gravity significantly alters dynamic processes controlled by a single mobile species, including colloidal aggregation and phase separation22 or multiscale self-assembly.23 In suspensions containing mobile constituents of varying size and mobility, gravity may play an even more significant role. Indeed, many physical processes are driven by differences in the dynamics of distinct constituents, such as suspension phase behavior,24–28 flow-induced margination,29, 30 or the self-organization of active matter.31–34 Studies of these processes in microgravity are expected to elucidate their complex physics; the varied nature of these systems requires a powerful, flexible, and easilyimplementable method, such as DDM. Although DDM has been extensively applied to systems featuring relatively simple dynamics described by a single characteristic relaxation time and to mixtures with Gaussian distribution of relaxation times such as protein aggregates, its application to systems featuring
1 Department of Chemical and Biomolecular Engineering, University of Houston, Houston, TX 77204-4004, USA Correspondence: Peter G. Vekilov ([email protected]) or Jacinta C. Conrad ([email protected])
Received: 1 March 2017 Revised: 10 June 2017 Accepted: 16 June 2017
Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA
Differential dynamic microscopy methods MS Safari et al.
2 nonuniform complex dynamics has been limited to samples with multi-step relaxations.35, 36 Here, we demonstrate a new application of DDM: the ability to resolve dynamics in a complex mixture containing two sizes of particles, using equipment comparable to that in the LMM on the ISS. We formulate dilute mixtures of polystyrene particles of radius 50 nm and 1 micron at different ratios of the large-to-small fraction at modest total volume fractions of ϕ ~ 10−3, at which both species freely diffuse. The 50 nm particles are too small to be resolved using standard optical methods. Using DLS and DDM, we measure the particle diffusivities in the mixtures. Whereas DLS is not sufficiently sensitive to resolve the dynamics of both species at these concentrations, DDM successfully measures the diffusivities of both large and small particles. The enhanced sensitivity of DDM derives from the preferential forward scattering of large objects. The scattered light from the large particles generates interference patterns that affect the amplitude of the dynamic correlation function. We show that this amplitude is nonmonotonic and corresponds to the interference pattern, and thus may be used to characterize their average axial position. We anticipate that this approach can be applied to time series of images acquired on the LMM and in other space experiments— enhancing the time resolution and providing new insights into microscale and nanoscale dynamics in microgravity. RESULTS AND DISCUSSION Dynamic light scattering As a control experiment, we measured the diffusivities of particles of radius 50 nm and 1 µm, respectively, using DLS. In suspensions containing particles of uniform size, the intermediate scattering function f(q,t) could be fitted to a single exponential, t (1) f ðq; tÞ ¼ exp τS where the time scale τS was related to the particle diffusivity via DS = 1/q2τS. The measured diffusivities of the small (4.3 ± 0.1 μm2 s−1) and large (0.20 ± 0.02 μm2 s−1) particles were in good agreement with the diffusivities predicted from the Stokes–Einstein equation using the nominal radii (4.3 μm2 s−1 and 0.21 μm2 s−1). To test the ability of DLS to measure dynamics of both species in a bidisperse mixture, we formulated samples containing a constant volume fraction of small particles, ϕS = 10−3, and added large particles at various concentrations to obtain volume fraction ratios of r = ϕL/ϕS = 0.03, 0.01, and 0.003. The intermediate scattering functions f(q,t), measured at three scattering angles, exhibited distinct shapes depending on the concentration of large particles. At the highest concentration of large particles (r = 0.03) and the lowest scattering wavevector (q = 6.8 µm−1), f(q,t) exhibited a second shoulder at long lag times (Fig. 1a); by contrast, no second shoulder was apparent at higher angles (e.g., for q = 18.7 µm−1 in Fig. 1a) or at lower concentrations of large particles (e.g., at q = 6.8 µm−1 and r = 0.003 in Fig. 1c). For bidisperse suspensions, the choice of an appropriate fitting model was determined by the scattering properties of the particles. The large particles used in these experiments were Mie scatterers37: the Mie parameter x for a particle of radius aL = 1 µm interacting with light of wavelength λ = 632.8 nm in water (refractive index n = 1.33) was x = 2πaLn/λ = 13.2, much larger than the Rayleigh threshold38 x = 1. The Mie parameter for the small particles was x = 0.66, slightly below this threshold. In the Mie regime, the scattering intensity is anisotropic with preferential forward scattering at low angles. Therefore, the contribution of both particles to the intermediate scattering function was angledependent and concentration-dependent. To capture these physics, we used two fitting forms: a single exponential decay when the scattering from small particles dominated and a double npj Microgravity (2017) 21
exponential decay when scattering from both populations was significant. For scattering experiments at r = 0.003 and 0.01, the correlation functions at all three scattering angles were fitted with a single exponential function (Eq. 1). The diffusivities calculated from the fitted time scale, DS = 1/q2τS, reflected the rate of diffusion of small particles over the length scale 2π/q and were in good agreement with that from the unary control experiment (Table 1). At a higher large-particle ratio r = 0.03, the intermediate correlation function exhibited a second shoulder at the lowest wavevector (q = 6.8 μm−1) indicative of two populations of diffusing particles. For r = 0.03, we fitted the intermediate correlation functions at q = 6.8 μm−1 to the sum of two exponential functions t t þ fL exp (2) f ðq; tÞ ¼ fS exp τS τL where τS and τL are the characteristic diffusion times of 50 nm and 1 µm particles, respectively, and fS and fL = 1−fS are proportional to the amplitude of the scattered signal produced by each particle population. Again, the diffusion coefficient for each particle species was extracted from its characteristic diffusion time via DS,L = 1/q2τS,L. The calculated diffusivities were larger than but comparable to those from the unary control measurements (Table 1); DL, in particular, was significantly larger. The inability to accurately detect the large particles across the accessible range of wavevectors prohibited the use of DLS to characterize minoritylarge bidisperse suspensions at low volume fractions. Hence, we explored alternate methods for characterizing dynamics in these samples. Differential dynamic microscopy To evaluate the sensitivity of DDM to distinguish particles of two different sizes, we performed DDM measurements on the same series of samples. In the DDM theory, the structure function Δ(q;t) is related to the intermediate scattering function f(q,t) via Δðq; tÞ ¼ AðqÞð1 f ðq; tÞÞ þ BðqÞ
(3)
where A(q) depends on the optical transfer function of the imaging system and on the scattering properties of objects, and B (q) captures any noise introduced into the system.9, 12 For a population of monodisperse scatterers at low concentration, f(q,t) is commonly fit to a single exponential decay (Eq. 1). In samples with more complex dynamics, such as those featuring multiple relaxation timescales,35, 36 a single exponential decay cannot be applied. Here, our goal was to determine the extent to which the sizes of particles in a bidisperse mixture could be resolved. Because DDM accesses a lower range of wavevectors than our DLS setup, the scattering intensity from the large particles is more pronounced than in DLS (Fig. 1d). Thus, we expected to observe two decays in the DDM signal for bidisperse mixtures, corresponding to the rate of diffusion for each particle size. At the highest concentration of large particles and at the lowest wavevectors, the signal from the large particles dominates; the small particles still contribute to the intensity at lower volume fraction ratios and higher wavevectors. To capture the contributions from both particles, we globally fit all relaxations across the wavevector range to the sum of two singleexponential functions with a weighting function fS(q) to describe the relative contribution from each particle population. This fitting form has five fitting parameters at each wavevector: τS(q), τL(q), A (q), B(q), and fS(q). To reduce the number of independent fitting parameters, we noted that the ratio of the decay rates of the large and small particles should be constant across the range of wavevectors, even as the relative scattering contribution from each was modulated by the anisotropic scattering of the particles. For bidisperse mixtures, we therefore implemented a global fitting
Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA
Differential dynamic microscopy methods MS Safari et al.
3
Fig. 1 a–c Intermediate scattering function f(q,t) as a function of lag time t measured for bidisperse mixtures of particles of radius 50 nm and 1 μm formulated at a large-to-small volume fraction ratio r of a 0.03, b 0.01, and c 0.003 at wavevectors of q = 6.8 μm−1 (30°, squares), 11.2 μm−1 (50°, diamonds), and 18.7 μm−1 (90°, triangles). Red lines indicate fitting functions: Eq. 2 for r = 0.03 and q = 6.8 μm−1 and Eq. 1 otherwise. d Predicted scattering intensity I(q) for small particles at ϕ = 10−3 and large particles at volume fraction ratios of r = 0.03, 0.01, and 0.003 as a function of wavevector q using standard equations for hard spheres.50 The range of wavevectors probed by DLS and DDM are indicated by dashed and dash-dotted lines, respectively. e–g Intermediate scattering function f(q,t), extracted from DDM measurements, as a function of lag time t measured for bidisperse mixtures of particles of radius 50 nm and 1 μm formulated at large-to-small volume fraction ratios r of a 0.03, b 0.01, and c 0.003. For each ratio, data were analyzed over the wavevector range 0.98 μm−1 < q < 3.01 μm−1; the figure shows representative correlation functions obtained for wavevectors q = 1.08 μm−1 (squares), 2.05 μm−1 (diamonds), or 2.92 μm−1 (triangles). Red lines indicate fits to Eq. 4
process and fit to the structure function Δðq; tÞ ¼ AðqÞ 1 fS ðqÞ exp
t τ L ðqÞ=fr
þ fL ðqÞ exp
t τ L ðqÞ
þ BðqÞ (4)
where τL(q) is the relaxation time of 1 µm particles at the wavevector q; the weighting functions fS(q) and fL(q) = 1−fS(q) describe the contribution of small (50 nm) and large (1 μm) particles, respectively, to the scattering intensity at q; and fr is the ratio of relaxation times of large and small particles, which is independent of q and hence was globally fit. Although the ratio fr is known for these particles from the control experiments, a priori knowledge of the particle sizes is not required to use this
Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA
npj Microgravity (2017) 21
Differential dynamic microscopy methods MS Safari et al.
4 Table 1. Diffusivities obtained from dynamic light scattering measurements for unary (top two rows) and bidisperse (labeled with volume fraction ratio r) samples Diffusion coefficient [μm2/s] q [μm−1] DS
Radius [nm] ϕ 50 r = ϕL/ϕS 0.003
1000 50 1000
0.01
50 1000
0.03
50 1000
DL
−3
4.3 ± 0.1
a
−5
a
0.20 ± 0.02
6.8
3.8 ± 0.2
b
11.2
4.1 ± 0.1
b
18.7
4.3 ± 0.1
b
6.8
4.5 ± 0.4
b
11.2
4.3 ± 0.1
b
18.7 6.8
4.5 ± 0.1 4.9 ± 0.3
b
3 × 10−5 11.2
4.1 ± 0.3
b
18.7
4.1 ± 0.1
b
10
10
10−3 −6
3 × 10 10−3
−5
1 × 10 10−3
0.30 ± 0.05
Error bars are the standard deviation from 10 independent runs. The Stokes-Einstein diffusivities are 4.3 and 0.21 μm2/s for small and large particles, respectively a measurements made on unary samples lacking this particle population b unable to resolve second particle population
functional form. This functional form exploits the full dynamic range of the DDM technique to generate a more robust non-linear fitting methodology and thereby accurately measure the diffusivities of both particles in a bidisperse mixture. We obtained f(q,t) for each wavevector from series of difference images.19 In contrast to the intermediate scattering functions measured at higher angles using DLS, the DDM f(q,t) clearly show non-exponential decays over 0.98 μm−1 < q < 3.01 μm−1 for all values of r examined here (Fig. 1e–g). This q-range is narrower than that accessed by us in earlier measurements using similar equipment13–15 and is limited by the dynamics of the particles relative to the rate of image acquisition (Fig. S1 in Supplementary Information). For q < 0.98 μm−1, the upper plateau was not reached by the maximum lag time at which we obtained enough independent measurements for statistics, 35 s (2200 frames), which was set by the frame rate and camera buffer. For q > 3.01 μm−1, the frame rate (63 fps) was insufficient to resolve the diffusive relaxation time scale of the smaller particles. Nonetheless, the data in Fig. 1e–g indicate that DDM can resolve particle dynamics in a bidisperse mixture. The intermediate plateaus observed in the DDM intermediate scattering functions (Fig. 1e–g) resulted from the large particle size ratio aaSL 20 . When aaSL
ARTICLE
OPEN
Differential dynamic microscopy of bidisperse colloidal suspensions Mohammad S. Safari1, Ryan Poling-Skutvik1, Peter G. Vekilov1 and Jacinta C. Conrad
1
Research tasks in microgravity include monitoring the dynamics of constituents of varying size and mobility in processes such as aggregation, phase separation, or self-assembly. We use differential dynamic microscopy, a method readily implemented with equipment available on the International Space Station, to simultaneously resolve the dynamics of particles of radius 50 nm and 1 μm in bidisperse aqueous suspensions. Whereas traditional dynamic light scattering fails to detect a signal from the larger particles at low concentrations, differential dynamic microscopy exhibits enhanced sensitivity in these conditions by accessing smaller wavevectors where scattering from the large particles is stronger. Interference patterns due to scattering from the large particles induce non-monotonic decay of the amplitude of the dynamic correlation function with the wavevector. We show that the position of the resulting minimum contains information on the vertical position of the particles. Together with the simple instrumental requirements, the enhanced sensitivity of differential dynamic microscopy makes it an appealing alternative to dynamic light scattering to characterize samples with complex dynamics. npj Microgravity (2017)3:21 ; doi:10.1038/s41526-017-0027-7
INTRODUCTION Microgravity provides a unique environment in which to investigate the physics of transport processes such as diffusion, convection, and conduction. These processes affect structure in systems featuring sub-microscale constituents, including bacterial biofilms,1, 2 protein crystals,3, 4 and complex fluids.5 Monitoring dynamics on these length scales in microgravity is expected to generate fundamental insight into the physics controlling structural evolution. One traditional method for characterizing dynamics at the microscale, dynamic light scattering (DLS),6 is already available on the International Space Station (ISS)7 but is restricted by the detector frame rate to characterize slow motions. An intriguing alternative is provided by recent enhancements to the Light Microscopy Module (LMM) on the ISS, which increased the time resolution of image acquisition and imparted confocal imaging capabilities. These advances make it possible to access faster dynamics across a broad range of samples but require methods to obtain dynamics from microscopy time series images. For microscale structures that can be directly visualized on an optical microscope, dynamics can be extracted from a time-series of microscopy images via image-processing algorithms. The numerical aperture on a standard light microscope, however, limits the size of resolvable structures to typically greater than 150 nm using the smallest feasible wavelength of light. Although advanced super-resolution microscopy methods can in principle lower this size limit significantly,8 low time resolution and stringent instrumentation requirements limit their immediate adoption on the ISS. Hence there remains a need for methods that can be readily implemented on a standard microscope with relatively simple equipment, compatible with the strict demands of the space environment. Differential dynamic microscopy (DDM),9 an extension of heterodyne near-field scattering10 and one of an emerging family
of digital Fourier techniques,11 is a flexible, powerful, and readilyimplemented method to probe microscale dynamics. In DDM, the microscale dynamics of a sample are extracted from the decorrelation of intensity fluctuations evaluated from a time series of difference images.12 This method has two key advantages: first, it has minimal instrumentation requirements, and, second, it can access smaller wavevectors and hence larger length scales than conventional DLS setups. Thus, DDM has been used to characterize the dynamics of dispersed nanoparticles13–15 and bacteria,16, 17 as well as colloidal18 and protein19 condensates. Further, DDM has been extended to imaging modes beyond brightfield, including fluorescence,12 confocal,20 and darkfield21 microscopy. Hence this method offers new flexibility and capability to investigate complex dynamic phenomena using microscopy. The simple instrumental requirements of DDM allow it to be implemented on the ISS to enable novel probes of dynamics in microgravity. As one example, gravity significantly alters dynamic processes controlled by a single mobile species, including colloidal aggregation and phase separation22 or multiscale self-assembly.23 In suspensions containing mobile constituents of varying size and mobility, gravity may play an even more significant role. Indeed, many physical processes are driven by differences in the dynamics of distinct constituents, such as suspension phase behavior,24–28 flow-induced margination,29, 30 or the self-organization of active matter.31–34 Studies of these processes in microgravity are expected to elucidate their complex physics; the varied nature of these systems requires a powerful, flexible, and easilyimplementable method, such as DDM. Although DDM has been extensively applied to systems featuring relatively simple dynamics described by a single characteristic relaxation time and to mixtures with Gaussian distribution of relaxation times such as protein aggregates, its application to systems featuring
1 Department of Chemical and Biomolecular Engineering, University of Houston, Houston, TX 77204-4004, USA Correspondence: Peter G. Vekilov ([email protected]) or Jacinta C. Conrad ([email protected])
Received: 1 March 2017 Revised: 10 June 2017 Accepted: 16 June 2017
Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA
Differential dynamic microscopy methods MS Safari et al.
2 nonuniform complex dynamics has been limited to samples with multi-step relaxations.35, 36 Here, we demonstrate a new application of DDM: the ability to resolve dynamics in a complex mixture containing two sizes of particles, using equipment comparable to that in the LMM on the ISS. We formulate dilute mixtures of polystyrene particles of radius 50 nm and 1 micron at different ratios of the large-to-small fraction at modest total volume fractions of ϕ ~ 10−3, at which both species freely diffuse. The 50 nm particles are too small to be resolved using standard optical methods. Using DLS and DDM, we measure the particle diffusivities in the mixtures. Whereas DLS is not sufficiently sensitive to resolve the dynamics of both species at these concentrations, DDM successfully measures the diffusivities of both large and small particles. The enhanced sensitivity of DDM derives from the preferential forward scattering of large objects. The scattered light from the large particles generates interference patterns that affect the amplitude of the dynamic correlation function. We show that this amplitude is nonmonotonic and corresponds to the interference pattern, and thus may be used to characterize their average axial position. We anticipate that this approach can be applied to time series of images acquired on the LMM and in other space experiments— enhancing the time resolution and providing new insights into microscale and nanoscale dynamics in microgravity. RESULTS AND DISCUSSION Dynamic light scattering As a control experiment, we measured the diffusivities of particles of radius 50 nm and 1 µm, respectively, using DLS. In suspensions containing particles of uniform size, the intermediate scattering function f(q,t) could be fitted to a single exponential, t (1) f ðq; tÞ ¼ exp τS where the time scale τS was related to the particle diffusivity via DS = 1/q2τS. The measured diffusivities of the small (4.3 ± 0.1 μm2 s−1) and large (0.20 ± 0.02 μm2 s−1) particles were in good agreement with the diffusivities predicted from the Stokes–Einstein equation using the nominal radii (4.3 μm2 s−1 and 0.21 μm2 s−1). To test the ability of DLS to measure dynamics of both species in a bidisperse mixture, we formulated samples containing a constant volume fraction of small particles, ϕS = 10−3, and added large particles at various concentrations to obtain volume fraction ratios of r = ϕL/ϕS = 0.03, 0.01, and 0.003. The intermediate scattering functions f(q,t), measured at three scattering angles, exhibited distinct shapes depending on the concentration of large particles. At the highest concentration of large particles (r = 0.03) and the lowest scattering wavevector (q = 6.8 µm−1), f(q,t) exhibited a second shoulder at long lag times (Fig. 1a); by contrast, no second shoulder was apparent at higher angles (e.g., for q = 18.7 µm−1 in Fig. 1a) or at lower concentrations of large particles (e.g., at q = 6.8 µm−1 and r = 0.003 in Fig. 1c). For bidisperse suspensions, the choice of an appropriate fitting model was determined by the scattering properties of the particles. The large particles used in these experiments were Mie scatterers37: the Mie parameter x for a particle of radius aL = 1 µm interacting with light of wavelength λ = 632.8 nm in water (refractive index n = 1.33) was x = 2πaLn/λ = 13.2, much larger than the Rayleigh threshold38 x = 1. The Mie parameter for the small particles was x = 0.66, slightly below this threshold. In the Mie regime, the scattering intensity is anisotropic with preferential forward scattering at low angles. Therefore, the contribution of both particles to the intermediate scattering function was angledependent and concentration-dependent. To capture these physics, we used two fitting forms: a single exponential decay when the scattering from small particles dominated and a double npj Microgravity (2017) 21
exponential decay when scattering from both populations was significant. For scattering experiments at r = 0.003 and 0.01, the correlation functions at all three scattering angles were fitted with a single exponential function (Eq. 1). The diffusivities calculated from the fitted time scale, DS = 1/q2τS, reflected the rate of diffusion of small particles over the length scale 2π/q and were in good agreement with that from the unary control experiment (Table 1). At a higher large-particle ratio r = 0.03, the intermediate correlation function exhibited a second shoulder at the lowest wavevector (q = 6.8 μm−1) indicative of two populations of diffusing particles. For r = 0.03, we fitted the intermediate correlation functions at q = 6.8 μm−1 to the sum of two exponential functions t t þ fL exp (2) f ðq; tÞ ¼ fS exp τS τL where τS and τL are the characteristic diffusion times of 50 nm and 1 µm particles, respectively, and fS and fL = 1−fS are proportional to the amplitude of the scattered signal produced by each particle population. Again, the diffusion coefficient for each particle species was extracted from its characteristic diffusion time via DS,L = 1/q2τS,L. The calculated diffusivities were larger than but comparable to those from the unary control measurements (Table 1); DL, in particular, was significantly larger. The inability to accurately detect the large particles across the accessible range of wavevectors prohibited the use of DLS to characterize minoritylarge bidisperse suspensions at low volume fractions. Hence, we explored alternate methods for characterizing dynamics in these samples. Differential dynamic microscopy To evaluate the sensitivity of DDM to distinguish particles of two different sizes, we performed DDM measurements on the same series of samples. In the DDM theory, the structure function Δ(q;t) is related to the intermediate scattering function f(q,t) via Δðq; tÞ ¼ AðqÞð1 f ðq; tÞÞ þ BðqÞ
(3)
where A(q) depends on the optical transfer function of the imaging system and on the scattering properties of objects, and B (q) captures any noise introduced into the system.9, 12 For a population of monodisperse scatterers at low concentration, f(q,t) is commonly fit to a single exponential decay (Eq. 1). In samples with more complex dynamics, such as those featuring multiple relaxation timescales,35, 36 a single exponential decay cannot be applied. Here, our goal was to determine the extent to which the sizes of particles in a bidisperse mixture could be resolved. Because DDM accesses a lower range of wavevectors than our DLS setup, the scattering intensity from the large particles is more pronounced than in DLS (Fig. 1d). Thus, we expected to observe two decays in the DDM signal for bidisperse mixtures, corresponding to the rate of diffusion for each particle size. At the highest concentration of large particles and at the lowest wavevectors, the signal from the large particles dominates; the small particles still contribute to the intensity at lower volume fraction ratios and higher wavevectors. To capture the contributions from both particles, we globally fit all relaxations across the wavevector range to the sum of two singleexponential functions with a weighting function fS(q) to describe the relative contribution from each particle population. This fitting form has five fitting parameters at each wavevector: τS(q), τL(q), A (q), B(q), and fS(q). To reduce the number of independent fitting parameters, we noted that the ratio of the decay rates of the large and small particles should be constant across the range of wavevectors, even as the relative scattering contribution from each was modulated by the anisotropic scattering of the particles. For bidisperse mixtures, we therefore implemented a global fitting
Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA
Differential dynamic microscopy methods MS Safari et al.
3
Fig. 1 a–c Intermediate scattering function f(q,t) as a function of lag time t measured for bidisperse mixtures of particles of radius 50 nm and 1 μm formulated at a large-to-small volume fraction ratio r of a 0.03, b 0.01, and c 0.003 at wavevectors of q = 6.8 μm−1 (30°, squares), 11.2 μm−1 (50°, diamonds), and 18.7 μm−1 (90°, triangles). Red lines indicate fitting functions: Eq. 2 for r = 0.03 and q = 6.8 μm−1 and Eq. 1 otherwise. d Predicted scattering intensity I(q) for small particles at ϕ = 10−3 and large particles at volume fraction ratios of r = 0.03, 0.01, and 0.003 as a function of wavevector q using standard equations for hard spheres.50 The range of wavevectors probed by DLS and DDM are indicated by dashed and dash-dotted lines, respectively. e–g Intermediate scattering function f(q,t), extracted from DDM measurements, as a function of lag time t measured for bidisperse mixtures of particles of radius 50 nm and 1 μm formulated at large-to-small volume fraction ratios r of a 0.03, b 0.01, and c 0.003. For each ratio, data were analyzed over the wavevector range 0.98 μm−1 < q < 3.01 μm−1; the figure shows representative correlation functions obtained for wavevectors q = 1.08 μm−1 (squares), 2.05 μm−1 (diamonds), or 2.92 μm−1 (triangles). Red lines indicate fits to Eq. 4
process and fit to the structure function Δðq; tÞ ¼ AðqÞ 1 fS ðqÞ exp
t τ L ðqÞ=fr
þ fL ðqÞ exp
t τ L ðqÞ
þ BðqÞ (4)
where τL(q) is the relaxation time of 1 µm particles at the wavevector q; the weighting functions fS(q) and fL(q) = 1−fS(q) describe the contribution of small (50 nm) and large (1 μm) particles, respectively, to the scattering intensity at q; and fr is the ratio of relaxation times of large and small particles, which is independent of q and hence was globally fit. Although the ratio fr is known for these particles from the control experiments, a priori knowledge of the particle sizes is not required to use this
Published in cooperation with the Biodesign Institute at Arizona State University, with the support of NASA
npj Microgravity (2017) 21
Differential dynamic microscopy methods MS Safari et al.
4 Table 1. Diffusivities obtained from dynamic light scattering measurements for unary (top two rows) and bidisperse (labeled with volume fraction ratio r) samples Diffusion coefficient [μm2/s] q [μm−1] DS
Radius [nm] ϕ 50 r = ϕL/ϕS 0.003
1000 50 1000
0.01
50 1000
0.03
50 1000
DL
−3
4.3 ± 0.1
a
−5
a
0.20 ± 0.02
6.8
3.8 ± 0.2
b
11.2
4.1 ± 0.1
b
18.7
4.3 ± 0.1
b
6.8
4.5 ± 0.4
b
11.2
4.3 ± 0.1
b
18.7 6.8
4.5 ± 0.1 4.9 ± 0.3
b
3 × 10−5 11.2
4.1 ± 0.3
b
18.7
4.1 ± 0.1
b
10
10
10−3 −6
3 × 10 10−3
−5
1 × 10 10−3
0.30 ± 0.05
Error bars are the standard deviation from 10 independent runs. The Stokes-Einstein diffusivities are 4.3 and 0.21 μm2/s for small and large particles, respectively a measurements made on unary samples lacking this particle population b unable to resolve second particle population
functional form. This functional form exploits the full dynamic range of the DDM technique to generate a more robust non-linear fitting methodology and thereby accurately measure the diffusivities of both particles in a bidisperse mixture. We obtained f(q,t) for each wavevector from series of difference images.19 In contrast to the intermediate scattering functions measured at higher angles using DLS, the DDM f(q,t) clearly show non-exponential decays over 0.98 μm−1 < q < 3.01 μm−1 for all values of r examined here (Fig. 1e–g). This q-range is narrower than that accessed by us in earlier measurements using similar equipment13–15 and is limited by the dynamics of the particles relative to the rate of image acquisition (Fig. S1 in Supplementary Information). For q < 0.98 μm−1, the upper plateau was not reached by the maximum lag time at which we obtained enough independent measurements for statistics, 35 s (2200 frames), which was set by the frame rate and camera buffer. For q > 3.01 μm−1, the frame rate (63 fps) was insufficient to resolve the diffusive relaxation time scale of the smaller particles. Nonetheless, the data in Fig. 1e–g indicate that DDM can resolve particle dynamics in a bidisperse mixture. The intermediate plateaus observed in the DDM intermediate scattering functions (Fig. 1e–g) resulted from the large particle size ratio aaSL 20 . When aaSL
